| L(s) = 1 | + (−1.28 − 1.52i)2-s + (1.23 − 2.73i)3-s + (−0.672 + 3.94i)4-s + (−8.74 − 1.74i)5-s + (−5.77 + 1.64i)6-s + (−1.59 + 3.84i)7-s + (6.89 − 4.05i)8-s + (−5.96 − 6.74i)9-s + (8.62 + 15.6i)10-s + (9.45 + 14.1i)11-s + (9.95 + 6.70i)12-s + (1.29 + 6.51i)13-s + (7.93 − 2.52i)14-s + (−15.5 + 21.7i)15-s + (−15.0 − 5.30i)16-s + (−0.662 − 0.662i)17-s + ⋯ |
| L(s) = 1 | + (−0.644 − 0.764i)2-s + (0.410 − 0.911i)3-s + (−0.168 + 0.985i)4-s + (−1.74 − 0.348i)5-s + (−0.961 + 0.273i)6-s + (−0.227 + 0.549i)7-s + (0.861 − 0.507i)8-s + (−0.662 − 0.749i)9-s + (0.862 + 1.56i)10-s + (0.859 + 1.28i)11-s + (0.829 + 0.558i)12-s + (0.0996 + 0.501i)13-s + (0.566 − 0.180i)14-s + (−1.03 + 1.45i)15-s + (−0.943 − 0.331i)16-s + (−0.0389 − 0.0389i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.425 - 0.904i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.425 - 0.904i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.233172 + 0.148022i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.233172 + 0.148022i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.28 + 1.52i)T \) |
| 3 | \( 1 + (-1.23 + 2.73i)T \) |
| good | 5 | \( 1 + (8.74 + 1.74i)T + (23.0 + 9.56i)T^{2} \) |
| 7 | \( 1 + (1.59 - 3.84i)T + (-34.6 - 34.6i)T^{2} \) |
| 11 | \( 1 + (-9.45 - 14.1i)T + (-46.3 + 111. i)T^{2} \) |
| 13 | \( 1 + (-1.29 - 6.51i)T + (-156. + 64.6i)T^{2} \) |
| 17 | \( 1 + (0.662 + 0.662i)T + 289iT^{2} \) |
| 19 | \( 1 + (9.76 - 1.94i)T + (333. - 138. i)T^{2} \) |
| 23 | \( 1 + (6.69 + 16.1i)T + (-374. + 374. i)T^{2} \) |
| 29 | \( 1 + (26.2 - 39.3i)T + (-321. - 776. i)T^{2} \) |
| 31 | \( 1 - 29.4iT - 961T^{2} \) |
| 37 | \( 1 + (56.9 + 11.3i)T + (1.26e3 + 523. i)T^{2} \) |
| 41 | \( 1 + (-3.42 - 8.27i)T + (-1.18e3 + 1.18e3i)T^{2} \) |
| 43 | \( 1 + (30.1 + 45.0i)T + (-707. + 1.70e3i)T^{2} \) |
| 47 | \( 1 + (-42.7 - 42.7i)T + 2.20e3iT^{2} \) |
| 53 | \( 1 + (18.6 + 27.8i)T + (-1.07e3 + 2.59e3i)T^{2} \) |
| 59 | \( 1 + (47.8 + 9.52i)T + (3.21e3 + 1.33e3i)T^{2} \) |
| 61 | \( 1 + (31.0 - 46.5i)T + (-1.42e3 - 3.43e3i)T^{2} \) |
| 67 | \( 1 + (12.0 - 18.0i)T + (-1.71e3 - 4.14e3i)T^{2} \) |
| 71 | \( 1 + (84.3 + 34.9i)T + (3.56e3 + 3.56e3i)T^{2} \) |
| 73 | \( 1 + (17.5 + 42.3i)T + (-3.76e3 + 3.76e3i)T^{2} \) |
| 79 | \( 1 + (85.6 + 85.6i)T + 6.24e3iT^{2} \) |
| 83 | \( 1 + (-1.45 - 7.31i)T + (-6.36e3 + 2.63e3i)T^{2} \) |
| 89 | \( 1 + (-2.46 + 5.94i)T + (-5.60e3 - 5.60e3i)T^{2} \) |
| 97 | \( 1 - 157. iT - 9.40e3T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.14039305677277445189751404739, −11.97627068706640239071671083343, −10.71508724540264694614941040422, −9.042158536867988765603893075510, −8.699934574112279436325083831759, −7.48536619668781491698431563498, −6.89034461979866664939685327154, −4.42069584868309890682357724969, −3.34781142763117372503640061084, −1.68377104691835056528629635935,
0.19198258157539474834911428104, 3.47946974709757672364266555432, 4.29244285577317956249764893005, 5.92140015105859553210683053579, 7.30220757892602037648882409471, 8.146405482348564152558913374128, 8.853381000588050650380390103411, 10.08776574588209532512219774505, 11.04901542863784516611644628067, 11.58914554908826181073062975929
